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      SUBROUTINE <a name="DLABRD.1"></a><a href="dlabrd.f.html#DLABRD.1">DLABRD</a>( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
     $                   LDY )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK auxiliary routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      INTEGER            LDA, LDX, LDY, M, N, NB
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAUP( * ),
     $                   TAUQ( * ), X( LDX, * ), Y( LDY, * )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="DLABRD.19"></a><a href="dlabrd.f.html#DLABRD.1">DLABRD</a> reduces the first NB rows and columns of a real general
</span><span class="comment">*</span><span class="comment">  m by n matrix A to upper or lower bidiagonal form by an orthogonal
</span><span class="comment">*</span><span class="comment">  transformation Q' * A * P, and returns the matrices X and Y which
</span><span class="comment">*</span><span class="comment">  are needed to apply the transformation to the unreduced part of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  If m &gt;= n, A is reduced to upper bidiagonal form; if m &lt; n, to lower
</span><span class="comment">*</span><span class="comment">  bidiagonal form.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  This is an auxiliary routine called by <a name="DGEBRD.27"></a><a href="dgebrd.f.html#DGEBRD.1">DGEBRD</a>
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  M       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The number of rows in the matrix A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  N       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The number of columns in the matrix A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  NB      (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The number of leading rows and columns of A to be reduced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
</span><span class="comment">*</span><span class="comment">          On entry, the m by n general matrix to be reduced.
</span><span class="comment">*</span><span class="comment">          On exit, the first NB rows and columns of the matrix are
</span><span class="comment">*</span><span class="comment">          overwritten; the rest of the array is unchanged.
</span><span class="comment">*</span><span class="comment">          If m &gt;= n, elements on and below the diagonal in the first NB
</span><span class="comment">*</span><span class="comment">            columns, with the array TAUQ, represent the orthogonal
</span><span class="comment">*</span><span class="comment">            matrix Q as a product of elementary reflectors; and
</span><span class="comment">*</span><span class="comment">            elements above the diagonal in the first NB rows, with the
</span><span class="comment">*</span><span class="comment">            array TAUP, represent the orthogonal matrix P as a product
</span><span class="comment">*</span><span class="comment">            of elementary reflectors.
</span><span class="comment">*</span><span class="comment">          If m &lt; n, elements below the diagonal in the first NB
</span><span class="comment">*</span><span class="comment">            columns, with the array TAUQ, represent the orthogonal
</span><span class="comment">*</span><span class="comment">            matrix Q as a product of elementary reflectors, and
</span><span class="comment">*</span><span class="comment">            elements on and above the diagonal in the first NB rows,
</span><span class="comment">*</span><span class="comment">            with the array TAUP, represent the orthogonal matrix P as
</span><span class="comment">*</span><span class="comment">            a product of elementary reflectors.
</span><span class="comment">*</span><span class="comment">          See Further Details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDA     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array A.  LDA &gt;= max(1,M).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  D       (output) DOUBLE PRECISION array, dimension (NB)
</span><span class="comment">*</span><span class="comment">          The diagonal elements of the first NB rows and columns of
</span><span class="comment">*</span><span class="comment">          the reduced matrix.  D(i) = A(i,i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  E       (output) DOUBLE PRECISION array, dimension (NB)
</span><span class="comment">*</span><span class="comment">          The off-diagonal elements of the first NB rows and columns of
</span><span class="comment">*</span><span class="comment">          the reduced matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  TAUQ    (output) DOUBLE PRECISION array dimension (NB)
</span><span class="comment">*</span><span class="comment">          The scalar factors of the elementary reflectors which
</span><span class="comment">*</span><span class="comment">          represent the orthogonal matrix Q. See Further Details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  TAUP    (output) DOUBLE PRECISION array, dimension (NB)
</span><span class="comment">*</span><span class="comment">          The scalar factors of the elementary reflectors which
</span><span class="comment">*</span><span class="comment">          represent the orthogonal matrix P. See Further Details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  X       (output) DOUBLE PRECISION array, dimension (LDX,NB)
</span><span class="comment">*</span><span class="comment">          The m-by-nb matrix X required to update the unreduced part
</span><span class="comment">*</span><span class="comment">          of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDX     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array X. LDX &gt;= M.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Y       (output) DOUBLE PRECISION array, dimension (LDY,NB)
</span><span class="comment">*</span><span class="comment">          The n-by-nb matrix Y required to update the unreduced part
</span><span class="comment">*</span><span class="comment">          of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDY     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array Y. LDY &gt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Further Details
</span><span class="comment">*</span><span class="comment">  ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  The matrices Q and P are represented as products of elementary
</span><span class="comment">*</span><span class="comment">  reflectors:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Each H(i) and G(i) has the form:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where tauq and taup are real scalars, and v and u are real vectors.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  If m &gt;= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
</span><span class="comment">*</span><span class="comment">  A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
</span><span class="comment">*</span><span class="comment">  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  If m &lt; n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
</span><span class="comment">*</span><span class="comment">  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
</span><span class="comment">*</span><span class="comment">  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  The elements of the vectors v and u together form the m-by-nb matrix
</span><span class="comment">*</span><span class="comment">  V and the nb-by-n matrix U' which are needed, with X and Y, to apply
</span><span class="comment">*</span><span class="comment">  the transformation to the unreduced part of the matrix, using a block
</span><span class="comment">*</span><span class="comment">  update of the form:  A := A - V*Y' - X*U'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  The contents of A on exit are illustrated by the following examples
</span><span class="comment">*</span><span class="comment">  with nb = 2:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  m = 6 and n = 5 (m &gt; n):          m = 5 and n = 6 (m &lt; n):
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
</span><span class="comment">*</span><span class="comment">    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
</span><span class="comment">*</span><span class="comment">    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
</span><span class="comment">*</span><span class="comment">    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
</span><span class="comment">*</span><span class="comment">    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
</span><span class="comment">*</span><span class="comment">    (  v1  v2  a   a   a  )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where a denotes an element of the original matrix which is unchanged,
</span><span class="comment">*</span><span class="comment">  vi denotes an element of the vector defining H(i), and ui an element
</span><span class="comment">*</span><span class="comment">  of the vector defining G(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Parameters ..
</span>      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      INTEGER            I
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Subroutines ..
</span>      EXTERNAL           DGEMV, <a name="DLARFG.145"></a><a href="dlarfg.f.html#DLARFG.1">DLARFG</a>, DSCAL
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          MIN
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Quick return if possible
</span><span class="comment">*</span><span class="comment">
</span>      IF( M.LE.0 .OR. N.LE.0 )
     $   RETURN
<span class="comment">*</span><span class="comment">
</span>      IF( M.GE.N ) THEN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Reduce to upper bidiagonal form
</span><span class="comment">*</span><span class="comment">
</span>         DO 10 I = 1, NB
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           Update A(i:m,i)
</span><span class="comment">*</span><span class="comment">
</span>            CALL DGEMV( <span class="string">'No transpose'</span>, M-I+1, I-1, -ONE, A( I, 1 ),
     $                  LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 )
            CALL DGEMV( <span class="string">'No transpose'</span>, M-I+1, I-1, -ONE, X( I, 1 ),
     $                  LDX, A( 1, I ), 1, ONE, A( I, I ), 1 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           Generate reflection Q(i) to annihilate A(i+1:m,i)
</span><span class="comment">*</span><span class="comment">
</span>            CALL <a name="DLARFG.172"></a><a href="dlarfg.f.html#DLARFG.1">DLARFG</a>( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
     $                   TAUQ( I ) )
            D( I ) = A( I, I )
            IF( I.LT.N ) THEN
               A( I, I ) = ONE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">              Compute Y(i+1:n,i)
</span><span class="comment">*</span><span class="comment">
</span>               CALL DGEMV( <span class="string">'Transpose'</span>, M-I+1, N-I, ONE, A( I, I+1 ),
     $                     LDA, A( I, I ), 1, ZERO, Y( I+1, I ), 1 )
               CALL DGEMV( <span class="string">'Transpose'</span>, M-I+1, I-1, ONE, A( I, 1 ), LDA,
     $                     A( I, I ), 1, ZERO, Y( 1, I ), 1 )
               CALL DGEMV( <span class="string">'No transpose'</span>, N-I, I-1, -ONE, Y( I+1, 1 ),
     $                     LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
               CALL DGEMV( <span class="string">'Transpose'</span>, M-I+1, I-1, ONE, X( I, 1 ), LDX,
     $                     A( I, I ), 1, ZERO, Y( 1, I ), 1 )
               CALL DGEMV( <span class="string">'Transpose'</span>, I-1, N-I, -ONE, A( 1, I+1 ),
     $                     LDA, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
               CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">              Update A(i,i+1:n)
</span><span class="comment">*</span><span class="comment">
</span>               CALL DGEMV( <span class="string">'No transpose'</span>, N-I, I, -ONE, Y( I+1, 1 ),
     $                     LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA )
               CALL DGEMV( <span class="string">'Transpose'</span>, I-1, N-I, -ONE, A( 1, I+1 ),
     $                     LDA, X( I, 1 ), LDX, ONE, A( I, I+1 ), LDA )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">              Generate reflection P(i) to annihilate A(i,i+2:n)
</span><span class="comment">*</span><span class="comment">
</span>               CALL <a name="DLARFG.201"></a><a href="dlarfg.f.html#DLARFG.1">DLARFG</a>( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
     $                      LDA, TAUP( I ) )
               E( I ) = A( I, I+1 )
               A( I, I+1 ) = ONE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">              Compute X(i+1:m,i)
</span><span class="comment">*</span><span class="comment">
</span>               CALL DGEMV( <span class="string">'No transpose'</span>, M-I, N-I, ONE, A( I+1, I+1 ),
     $                     LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 )
               CALL DGEMV( <span class="string">'Transpose'</span>, N-I, I, ONE, Y( I+1, 1 ), LDY,
     $                     A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
               CALL DGEMV( <span class="string">'No transpose'</span>, M-I, I, -ONE, A( I+1, 1 ),
     $                     LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
               CALL DGEMV( <span class="string">'No transpose'</span>, I-1, N-I, ONE, A( 1, I+1 ),
     $                     LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
               CALL DGEMV( <span class="string">'No transpose'</span>, M-I, I-1, -ONE, X( I+1, 1 ),
     $                     LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
               CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
            END IF
   10    CONTINUE
      ELSE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Reduce to lower bidiagonal form
</span><span class="comment">*</span><span class="comment">
</span>         DO 20 I = 1, NB
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           Update A(i,i:n)
</span><span class="comment">*</span><span class="comment">
</span>            CALL DGEMV( <span class="string">'No transpose'</span>, N-I+1, I-1, -ONE, Y( I, 1 ),
     $                  LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA )
            CALL DGEMV( <span class="string">'Transpose'</span>, I-1, N-I+1, -ONE, A( 1, I ), LDA,
     $                  X( I, 1 ), LDX, ONE, A( I, I ), LDA )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           Generate reflection P(i) to annihilate A(i,i+1:n)
</span><span class="comment">*</span><span class="comment">
</span>            CALL <a name="DLARFG.236"></a><a href="dlarfg.f.html#DLARFG.1">DLARFG</a>( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
     $                   TAUP( I ) )
            D( I ) = A( I, I )
            IF( I.LT.M ) THEN
               A( I, I ) = ONE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">              Compute X(i+1:m,i)
</span><span class="comment">*</span><span class="comment">
</span>               CALL DGEMV( <span class="string">'No transpose'</span>, M-I, N-I+1, ONE, A( I+1, I ),
     $                     LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 )
               CALL DGEMV( <span class="string">'Transpose'</span>, N-I+1, I-1, ONE, Y( I, 1 ), LDY,
     $                     A( I, I ), LDA, ZERO, X( 1, I ), 1 )
               CALL DGEMV( <span class="string">'No transpose'</span>, M-I, I-1, -ONE, A( I+1, 1 ),
     $                     LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
               CALL DGEMV( <span class="string">'No transpose'</span>, I-1, N-I+1, ONE, A( 1, I ),
     $                     LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 )
               CALL DGEMV( <span class="string">'No transpose'</span>, M-I, I-1, -ONE, X( I+1, 1 ),
     $                     LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
               CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">              Update A(i+1:m,i)
</span><span class="comment">*</span><span class="comment">
</span>               CALL DGEMV( <span class="string">'No transpose'</span>, M-I, I-1, -ONE, A( I+1, 1 ),
     $                     LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 )
               CALL DGEMV( <span class="string">'No transpose'</span>, M-I, I, -ONE, X( I+1, 1 ),
     $                     LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">              Generate reflection Q(i) to annihilate A(i+2:m,i)
</span><span class="comment">*</span><span class="comment">
</span>               CALL <a name="DLARFG.265"></a><a href="dlarfg.f.html#DLARFG.1">DLARFG</a>( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1,
     $                      TAUQ( I ) )
               E( I ) = A( I+1, I )
               A( I+1, I ) = ONE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">              Compute Y(i+1:n,i)
</span><span class="comment">*</span><span class="comment">
</span>               CALL DGEMV( <span class="string">'Transpose'</span>, M-I, N-I, ONE, A( I+1, I+1 ),
     $                     LDA, A( I+1, I ), 1, ZERO, Y( I+1, I ), 1 )
               CALL DGEMV( <span class="string">'Transpose'</span>, M-I, I-1, ONE, A( I+1, 1 ), LDA,
     $                     A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
               CALL DGEMV( <span class="string">'No transpose'</span>, N-I, I-1, -ONE, Y( I+1, 1 ),
     $                     LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
               CALL DGEMV( <span class="string">'Transpose'</span>, M-I, I, ONE, X( I+1, 1 ), LDX,
     $                     A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
               CALL DGEMV( <span class="string">'Transpose'</span>, I, N-I, -ONE, A( 1, I+1 ), LDA,
     $                     Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
               CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
            END IF
   20    CONTINUE
      END IF
      RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     End of <a name="DLABRD.288"></a><a href="dlabrd.f.html#DLABRD.1">DLABRD</a>
</span><span class="comment">*</span><span class="comment">
</span>      END

</pre>

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